Geometry Through Art

Norman Shapiro

Sample Lesson - The Square

Table of Contents

I. The Square

A. Coloring Squares
3. Exploring numbers that yield squares and rectangles

When I give this lesson during an artist-in-residency presentation, the children are seated in twos or fours and supplied with 2-4 colors to be shared. I am at the front of the room with my overhead projector and some felt tip colored markers to use on a transparency showing the square grid.

I ask the children to color in one of the smallest squares on the grid. I tell them they can put that square anywhere they like on the paper. I say,

"Now that's a 1-square square.

"Let's take another color marker and color in a bigger square. We don't want it to touch our 1-square square. How many squares must we color in to make a bigger square? Don't tell me; show me."

I walk around and look at the squares that are colored in. I will see squares that have 4 squares colored in, 9 squares, 16, etc. I will also see rectangles.

"Boys and girls, let's make the squares in size order. After the 1-square square, what is the next size square?"

To facilitate the exploration of 'square numbers' I will use poker chips on the overhead projector and arrange 4 of them 'to make a square'. I will ask a student to come up to the overhead and add poker chips to make the 4-square into a bigger square. Before the student does each successive size, I will ask, "how many must we add?"

On the blackboard, two charts are being built as we go along:

      
  Sequence      # of Squares       Add    Yields alternating
 of Squares     in the Square    Squares    odd and even #s

 1st square      1-sq. square        3          even
 2nd square      4-sq. square        5          odd
 3rd square      9-sq. square        7          even
 4th square     16-sq. square        9          odd
 5th square     25-sq. square       11          even
 6th square     36-sq. square       13          odd
 7th square     49-sq. square       15          even
 8th square     64-sq. square            etc.
 9th square     81-sq. square
10th square    100-sq. square

Thus there are patterns of odd and even to help us know if the number of unit squares we think must be added to form a larger composite square is right, and the nth row of the 'Add Squares' column is equal to the number of squares you have to add to form the nth square from the n-1st one.

As we color in the successive unit squares we see a pattern on the grid. Each bigger composite square increases in width and height by 1. We count one more horizontal square, and a corresponding increase in vertical squares. The 1st square has 1 square, the 2nd has 2 squares, the 3rd 3, and, so on.

Unlike squares, rectangles have different numbers of squares in the vertical direction than the horizontal direction. Numbers that yield squares also yield rectangles: 9, for instance, makes a 3x3 square and a 9x1 rectangle.

Students who are members of the 'Secret Square Society' know and can give me upon my asking a 'square number'. They will be able to visualize a rectangle by the number! They will know that certain numbers can be made into more than one rectangle. Children enjoy being in on the secret - and the game doesn't end until everyone gets into the Secret Square Society.

I point out to teachers that the numbers that do not make a rectangle or a square on the grid are prime numbers.

Coloring square grids can be very useful in:

• developing one's powers to visualize number patterns,
• understanding factoring and multiplication,
• developing concepts of perimeter and area,
• identifying prime numbers,
• doing everyday arithmetic operations.

Square grids make math artful. Students can make designs with squares that become

• letters of the alphabet,
• pictures,
• regular repeat patterns,
• original designs for needlepoint,
and much more.

Students who are more visual than verbal may find these experiences especially meaningful.

On to Classroom Supplies You Will Need

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3 November 1995